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❆❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ♦✈❡r ❢r❡❡ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞

❆rt❡♠ ❙❤❡✈❧②❛❦♦✈ ✸✵t❤ ❆♣r✐❧

■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥

▲❡t A∗ ❜❡ t❤❡ ❢r❡❡ s❡♠✐❣r♦✉♣ ♦❢ ❛♥ ❛❧♣❤❛❜❡t A = {❛❧❧ ❘✉ss✐❛♥ ❧❡tt❡rs}✳ ❆♥② ❘✉ss✐❛♥ ✇♦r❞ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ A∗✳ ▲❡t A∗

1 ⊆ A∗ ❜❡ t❤❡ s❡t ♦❢ ✇♦r❞s

✇✐t❤ ♥♦ r❡♣❡t✐t✐♦♥s✳ ■s t❤❡ s❡t A∗

1 ✐♥t❡r❡st✐♥❣❄

▼❛♥② ♣♦♣✉❧❛r ✭s❡❡ ❚❱✮ ✇♦r❞s ❞♦ ♥♦t ❝♦♥t❛✐♥ ❧❡tt❡r r❡♣❡t✐t✐♦♥s✿ Ïóòèí✱ âîäêà✱ êðûìíàø✱ Ñòàëèí✱ íåôòü✱ áàíäåðîâöû✱ ✃àäûðîâ✱ ðóáëü✱ äåìîêðàòèÿ✱ ➹åéðîïà✱ äóõîâíûý ñêðåïû✱ âàòíèê✳

❆♠❛③✐♥❣❧②✱ t❤❡r❡ ❡①✐st s❡♠✐❣r♦✉♣s t❤❛t ❞❡s❝r✐❜❡ ✇♦r❞s ✇✐t❤ ♥♦ ❧❡tt❡r r❡♣❡t✐t✐♦♥s✳ ❆ s❡♠✐❣r♦✉♣ ✐s ❝❛❧❧❡❞ ❛ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞ ✭▲❘❇✮ ✐❢ t❤❡ ✐❞❡♥t✐t✐❡s xx = x✱ xyx = xy ❤♦❧❞✳

❉❡✜♥✐t✐♦♥

❚❤❡ ❢r❡❡ ▲❘❇ Fn ♦❢ r❛♥❦ n ✐s t❤❡ s❡t ♦❢ ❛❧❧ ✇♦r❞s ♦❢ t❤❡ ❛❧♣❤❛❜❡t {a1, a2, . . . , an} s✉❝❤ t❤❛t ❡❛❝❤ ✇♦r❞ ❝♦♥s✐sts ♦❢ ❞✐✛❡r❡♥t ❧❡tt❡rs✳ ❊✳❣✳ F3 = {a1, a2, a3, a1a2, a2a1, a1a3, a3a1, a3a2, a2a3, a1a2a3, a1a3a2, a2a1a3, a2a3a1, a3a1a2, a3a2a1} ❚❤❡ ♣r♦❞✉❝t ♦❢ w1, w2 ∈ Fn ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ w1 ◦ w2 = w1(w2)∃, ✇❤❡r❡ t❤❡ ♦♣❡r❛t♦r ∃ ❞❡❧❡t❡s ❛❧❧ ❧❡tt❡rs ♦❢ w2 ✇❤✐❝❤ ♦❝❝✉r ❡❛r❧✐❡r✳ ❋♦r ❡①❛♠♣❧❡✱ (a1a2)(a2a3a1) = a1a2a3. ❖❜✈✐♦✉s❧②✱ ❡❧❡♠❡♥ts ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ❧❡tt❡rs ❛r❡ ❧❡❢t ③❡r♦❡s✿ (a1a2a3)x = a1a2a3 ❢♦r ❛❧❧ x.

P❛rt✐❛❧ ♦r❞❡rs ♦✈❡r ▲❘❇✲s

x ≤ y ⇔ xy = y ❋♦r ❡❧❡♠❡♥ts ♦❢ Fn x ≤ y ♠❡❛♥s t❤❛t x ✐s ❛ ♣r❡✜① ♦❢ y✳ ❊✳❣✳ a1a2 ≤ a1a2a3, a1 ≤ a1. ≤✲❝♦♠♣❛r❛❜❧❡ ❡❧❡♠❡♥ts ❛r❡ ❛❧✇❛②s ❝♦♠♠✉t❡✳

≤✲♦r❞❡r ♦✈❡r Fn ✐s ❛ tr❡❡

▲❡t ✉s ❛❞❥♦✐♥t ❛♥ ✐❞❡♥t✐t② ε t♦ Fn✱ ❤❡♥❝❡ t❤❡ ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ ≤ ✐s ✏✏✏✏✏✏✏✏✏✏✏✏ ❅ ❅ ❅ ❅ ❅ ✁ ✁ ✁ ✁ ε ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ a2 ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ a3 ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ a1 a1a2 a1a3 a3a1 a3a2 a2a1 a2a3 a1a2a3 a1a3a2 a2a1a3 a2a3a1 a3a1a2 a3a2a1

❘❛♥❞♦♠ ✇❛❧❦s

❚s❡t❧✐♥ ❧✐❜r❛r②✿ ❛ s❤❡❧❢ ♦❢ ❜♦♦❦s✳ ❆♥② ❜♦♦❦ ❤❛s ❛ ♣r♦❜❛❜✐❧✐t② pi✳ ❲✐t❤ t❤❡ ❣✐✈❡♥ ♣r♦❜❛❜✐❧✐t✐❡s ✇❡ ❝❤♦♦s❡ ❛ ❜♦♦❦ ❛♥❞ ♣✉t ✐t ❛t t❤❡ ❢r♦♥t ♦❢ t❤❡ s❤❡❧❢✳ ■♥ ✐s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❢r❡❡ ▲❘❇ Fn✱ ✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜♦♦❦s✳ ❚❤❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ✇❛❧❦ ✐s ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥s✳ ❙❡❡ ❑✳ ❙✳ ❇r♦✇♥✱ ❙❡♠✐❣r♦✉♣s✱ r✐♥❣s✱ ❛♥❞ ▼❛r❦♦✈ ❝❤❛✐♥s✱ ❏✳ ❚❤❡♦r❡t✳ Pr♦❜❛❜✳✱ ✭✷✵✵✵✮✱ ✶✸✭✸✮✱ ✽✼✶✕✾✸✽✳ ❢♦r ❛♥♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ▲❘❇ ✐♥ r❛♥❞♦♠ ✇❛❧❦s✳

▼♦t✐✈❛t✐♦♥✿ ♠❛tr♦✐❞s

❆ ♠❛tr♦✐❞ M ✐s ❛ ♣❛✐r (E, I)✱ ✇❤❡r❡ E ✐s ❛ s❡t ✭❝❛❧❧❡❞ t❤❡ ❣r♦✉♥❞ s❡t✮ ❛♥❞ I ✐s ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ E ✭❝❛❧❧❡❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥t s❡ts✮ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✶ ❊✈❡r② s✉❜s❡t ♦❢ ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t ✐s ✐♥❞❡♣❡♥❞❡♥t✱ ✐✳❡✳✱ ❢♦r ❡❛❝❤

A′ ⊂ A ⊂ E✱ ✐❢ A ∈ I t❤❡♥ A′ ∈ I ✭❤❡r❡❞✐t❛r② ♣r♦♣❡rt②✮✳

✷ ■❢ A ❛♥❞ B ❛r❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ I ❛♥❞ |A| > |B|✱ t❤❡♥ t❤❡r❡

❡①✐sts ❛♥ ❡❧❡♠❡♥t ✐♥ A t❤❛t ✇❤❡♥ ❛❞❞❡❞ t♦ B ❣✐✈❡s ❛ ❧❛r❣❡r ✐♥❞❡♣❡♥❞❡♥t s❡t t❤❛♥ B ✭❡①❝❤❛♥❣❡ ♣r♦♣❡rt②✮✳ ❚❛❦❡ t❤r❡❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦rs E = {v1, v2, v3} ♦❢ s♦♠❡ ✈❡❝t♦r s♣❛❝❡✳ ❚❤❡♥ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ t❤❡ ♠❛tr♦✐❞ M = (E, I) ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ∅, {v1}, {v2}, {v3}, {v1, v2}, {v1, v3}, {v2, v3}, {v1, v2, v3}

❚❤❡ ❢r❡❡ ♠❛tr♦✐❞ ♦❢ r❛♥❦ n ✐s t❤❡ s✐♠♣❧❡st ♠❛tr♦✐❞ ❡✈❡r✳ ■t ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ✐♥❞❡♣❡♥❞❡♥t s❡ts ❣❡♥❡r❛t❡❞ ❜② n ✈❡❝t♦rs✳ Pr♦❜❧❡♠ ❆r❡ t❤❡r❡ ♦t❤❡r s✐♠♣❧❡ ♠❛tr♦✐❞s❄ ■❞❡❛✦ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r ♠❛tr♦✐❞s ❛s ▲❘❇✲s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ♠❛tr♦✐❞s✳

▲❡t M = (E, I) ❜❡ ❛ ♠❛tr♦✐❞ ❛♥❞ − → I ❜❡ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ♦r❞❡r❡❞ s❡ts ♦❢ I✳ ❉❡✜♥❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r − → I ❜② (v1, v2, . . . , vn)(u1, u2, . . . , um) = (v1, v2, . . . , vn, ui1, ui2, . . . , uik), ✇❤❡r❡ ❡❛❝❤ uij ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r❡✈✐♦✉s ❡❧❡♠❡♥ts✳ ❚❤✉s✱ t❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ✐♥❞❡♣❡♥❞❡♥t s❡ts − → I ❜❡❝♦♠❡s ❛♥ ▲❘❇✳ ❋♦r ❡①❛♠♣❧❡✱ − → I ♦❢ t❤❡ ♠❛tr♦✐❞ ❣❡♥❡r❛t❡❞ ❜② ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦rs v1, v2, v3 ✐s ✐s♦♠♦r♣❤✐❝ t♦ F3✳

▲❡t − → I n ❜❡ ❛❧❧ ♦r❞❡r❡❞ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ t❤❡ ❢r❡❡ ♠❛tr♦✐❞ ♦❢ r❛♥❦ n✳ − → I n ✐s ✐s♦♠♦r♣❤✐❝ t♦ Fn✳ ❉❡♥♦t❡ t❤❡ ❢r❡❡ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞ ♦❢ ❝♦✉♥t❛❜❧❡ r❛♥❦ ❜② F ✭✐t ❝♦♥t❛✐♥s ❛❧❧ Fn✮✳ ▲❡t ✉s ❣✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥✿ ✏❛ ✜♥✐t❡ ♠❛tr♦✐❞ M = (E, I) ✐s s✐♠♣❧❡ ✐❢ − → I ✐s ❡♠❜❡❞❞❡❞ ✐♥ F✑✳ ❚❤✉s✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ st✉❞② ✜♥✐t❡ s✉❜❜❛♥❞s ♦❢ F✳

Pr♦♣❡rt✐❡s ♦❢ s✉❜❜❛♥❞s ✐♥ F

❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♦r❞❡r ≤ ✐s ❛ tr❡❡ ❢♦r ❛♥② s✉❜❜❛♥❞ S ♦❢ F✳ ❇❛♥❞s ✇✐t❤ s✉❝❤ ♣r♦♣❡rt② ♦❢ ≤ ❛r❡ ❝❛❧❧❡❞ r✐❣❤t ❤❡r❡❞✐t❛r②✳ ❆r❡ ♦t❤❡r ♣r♦♣❡rt✐❡s❄ ▲❡t α(x) ❜❡ t❤❡ ❛♥❝❡st♦r ♦❢ x ∈ S r❡❧❛t✐✈❡ t❤❡ ♦r❞❡r ≤✳ ❙✐♥❝❡ t❤❡ ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ ≤ ✐s ❛ tr❡❡✱ α(x) ✐s ✉♥✐q✉❡✳ Sc = {s | α(s)c = sc, α(s)α(c) = sα(c)}. ❲❡ s❛② t❤❛t S ❤❛s ❧♦❝❛❧ ❧✐♥❡❛r ♦r❞❡r ✐❢ ❡❛❝❤ s❡t Sc ❛❞♠✐ts ❛ ❧✐♥❡❛r ♦r❞❡r ⊏c s✉❝❤ t❤❛t

✶ ✐❢ Sb ⊆ Sc ❛♥❞ x ⊏b y t❤❡♥ x ⊏c y❀ ✷ ✈❡r② ❝♦♠♣❧✐❝❛t❡❞ ❝♦♥❞✐t✐♦♥❀ ✸ ❜♦r✐♥❣ ❝♦♥❞✐t✐♦♥✳

Pr♦♣❡rt✐❡s ♦❢ s✉❜❜❛♥❞s ✐♥ F

❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♦r❞❡r ≤ ✐s ❛ tr❡❡ ❢♦r ❛♥② s✉❜❜❛♥❞ S ♦❢ F✳ ❇❛♥❞s ✇✐t❤ s✉❝❤ ♣r♦♣❡rt② ♦❢ ≤ ❛r❡ ❝❛❧❧❡❞ r✐❣❤t ❤❡r❡❞✐t❛r②✳ ❆r❡ ♦t❤❡r ♣r♦♣❡rt✐❡s❄ ▲❡t α(x) ❜❡ t❤❡ ❛♥❝❡st♦r ♦❢ x ∈ S r❡❧❛t✐✈❡ t❤❡ ♦r❞❡r ≤✳ ❙✐♥❝❡ t❤❡ ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ ≤ ✐s ❛ tr❡❡✱ α(x) ✐s ✉♥✐q✉❡✳ Sc = {s | α(s)c = sc, α(s)α(c) = sα(c)}. ❲❡ s❛② t❤❛t S ❤❛s ❧♦❝❛❧ ❧✐♥❡❛r ♦r❞❡r ✐❢ ❡❛❝❤ s❡t Sc ❛❞♠✐ts ❛ ❧✐♥❡❛r ♦r❞❡r ⊏c s✉❝❤ t❤❛t

✶ ✐❢ Sb ⊆ Sc ❛♥❞ x ⊏b y t❤❡♥ x ⊏c y❀ ✷ ✈❡r② ❝♦♠♣❧✐❝❛t❡❞ ❝♦♥❞✐t✐♦♥❀ ✸ ❜♦r✐♥❣ ❝♦♥❞✐t✐♦♥✳

❚❤❡♦r❡♠ ❋♦r ❛ ✜♥✐t❡ ▲❘❇ S t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✶ S ✐s ❡♠❜❡❞❞❡❞ ✐♥t♦ F❀ ✷ S ✐s r✐❣❤t ❤❡r❡❞✐t❛r② ❛♥❞ ❤❛s ❛ ❧♦❝❛❧ ❧✐♥❡❛r ♦r❞❡r✳ ✸ S ✐s ❛ ❝♦♦r❞✐♥❛t❡ ❜❛♥❞ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❛❧❣❡❜r❛✐❝ s❡t ♦✈❡r F

✭❡q✉❛t✐♦♥s ✇✐t❤ ♥♦ ❝♦♥st❛♥ts✮❀

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▲❡t S ❜❡ ❛♥ ▲❘❇✳ ❈♦♥s✐❞❡r ❛ ❝♦♥❣r✉❡♥❝❡ θ✿ x ∼ y ⇔ xy = x, yx = y. ❚❤❡ q✉♦t✐❡♥t ♠❛♣ σ: S → S/θ ♣r♦❞✉❝❡s ❛ s❡♠✐❧❛tt✐❝❡ σ(S) ✭✐✳❡✳ σ(S) ✐s ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ ✐❞❡♠♣♦t❡♥t✮✳ ■♥ t❤❡ s❡q✉❡❧ ✇❡ ❞❡♥♦t❡ s❡♠✐❧❛tt✐❝❡ ❡❧❡♠❡♥ts ❜② ❜♦❧❞ ❧❡tt❡rs✳ ▲❡t F ❜❡ ❛ ❢r❡❡ s❡♠✐❧❛tt✐❝❡ ♦❢ ❝♦✉♥t❛❜❧❡ r❛♥❦✳ ❖❜✈✐♦✉s❧②✱ σ(F) = F✳ ❋r❡❡ ❣❡♥❡r❛t♦rs ♦❢ F ✭F✮ ❛r❡ ❞❡♥♦t❡❞ ❜② a1, a2, . . . ✭a1, a2, . . .✮ ❛♥❞ ✇❡ ❤❛✈❡ σ(ai) = ai✳

❍♦✇ t♦ ❞❡✜♥❡ ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ S ✐♥t♦ F❄

❇❡❧♦✇ ✇❡ ❞❡✜♥❡ ❛♥ ❡♠❜❡❞❞✐♥❣ λ: S → F✳

✶ ❚❛❦❡ s ∈ S❀ ✷ ❣❡t σ(s) ∈ σ(S)❀ ✸ ❡♠❜❡❞ σ(S) ✐♥t♦ F❀ ✹ ♥♦✇ σ(s) = a1a2a3❀ ✺ ✉s✐♥❣ ❧♦❝❛❧ ❧✐♥❡❛r ♦r❞❡r✱ s♦rt ❧❡tt❡rs ai❀ ❢♦r ❡①❛♠♣❧❡✱ σ(s) = a2a3a1❀ ✻ r❡♣❧❛❝❡ ❡❧❡♠❡♥ts ai ∈ F t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❣❡♥❡r❛t♦rs ai ∈ F ❛♥❞

♦❜t❛✐♥

✼ λ(s) = a2a3a1 ∈ F✳